Problem 41 - College Algebra with Modeling and Visualization

Question

Find a general term \(a_{n}\) for the arithmetic sequence. $$a_{1}=5, d=-2$$

Step-by-Step Solution

Answer

The general term is \(a_n = 7 - 2n\).

Step 1: Understand the Arithmetic Sequence Components

An arithmetic sequence is a sequence of numbers with a constant difference between consecutive terms. Here, you are given the first term of the sequence, \(a_1 = 5\), and the common difference, \(d = -2\). The aim is to find the formula for the general term \(a_n\).

Step 2: Recall the Formula for the General Term

The general term \(a_n\) of an arithmetic sequence can be calculated using the formula: \[ a_n = a_1 + (n-1) \, d \] where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.

Step 3: Substitute Given Values into the Formula

Using the values provided in the exercise, substitute \(a_1 = 5\) and \(d = -2\) into the general term formula: \[ a_n = 5 + (n-1) \, (-2) \]

Step 4: Simplify the Expression

Simplify the expression \[ a_n = 5 + (n-1)(-2) \] by distributing \(-2\) and combining like terms: \[ a_n = 5 - 2(n-1) \] \[ a_n = 5 - 2n + 2 \] \[ a_n = 7 - 2n \]

Step 5: Express the Final General Term

The simplified expression for the general term of the sequence is: \[ a_n = 7 - 2n \] This is the formula to find any term in the arithmetic sequence.

Key Concepts

General Term Formula

The general term formula is a fundamental concept in understanding arithmetic sequences. It allows you to find any term within the sequence if the sequence has a consistent pattern, known as the common difference.

For an arithmetic sequence, the formula for the general term \( a_n \) is given by:

• \( a_n = a_1 + (n-1) \times d \)

Here, \( a_1 \) represents the first term of the sequence, \( n \) indicates the position of the term you want to find, and \( d \) is the common difference.

The purpose of this formula is to calculate any term number \( n \) without listing all preceding terms. It simplifies the task and makes handling arithmetic sequences more straightforward. For our example, by substituting the given values, this formula lets us find the general rule of the sequence.

Common Difference

The common difference in an arithmetic sequence is the numerical gap between consecutive terms. This is a crucial element because it determines the uniform pattern that the sequence follows.

In more formal terms, the common difference \( d \) is calculated by subtracting any term from its succeeding term:

• \( d = a_{n+1} - a_n \)

For example, if you look at the sequence with \( a_1 = 5 \) and \( d = -2 \), every next term reduces by 2. This consistent reduction or addition is what makes the sequence arithmetic, allowing us to use the general term formula.

Remember, if the common difference is positive, the sequence will increase, while a negative common difference means the sequence decreases progressively.

Sequence Simplification

Sequence simplification involves expressing the sequence in its simplest form, making it more practical to work with and understand.

After substituting the values into the general term formula, you end up with an expression that might seem complex.
In our example, after substituting \( a_1 = 5 \) and \( d = -2 \), we managed to simplify the equation to:

• \( a_n = 7 - 2n \)

This step involves distributing the common difference and combining like terms.

Simplifying arithmetic sequences helps in identifying patterns and easily computing any term \( a_n \). It reduces the cognitive load needed to solve problems and, ultimately, provides a clearer understanding of the sequence's structure.